What I'm about to say is possibly less strictly correct, but here's the gist of what I'm getting. Text hedge incoming (not quite a text wall).
But if you've got two sets A and B, then you have another set, A x B, which is made of ordered pairs from those. Specifically, the first item in the pair comes from one set, and the second from the other. For instance, take coordinates in a plane - however you write it, you never have two x coordinates or two y coordinates. You always have one of each. So we'll call the item we get from A, whatever it is, a, and likewise the item from B is b. And A x B is a set containing every possible pair of values that meet that requirement.
In order for something to be a function, it needs to be a subset of A x B, which is to say every item in the function must also be in A x B. Also, any pairs that have a given a value must share a b value (or, in other words, there is one and only one possible b corresponding to a given a). For instance, if we look at coordinates in a plane again, we're using every possible coordinate along the x axis as A, and every possible coordinate along the y axis as our B. If we have something we think is a function, we check to make sure that it's really a set of coordinate pairs (which equations work out to being, in a sort of implicit way, I think), that all of those pairs consist of an x and a y coordinate, and that every x coordinate that IS part of the set* has one, and only one, y coordinate.
Note that we're distinguishing between A and B, in that while a given a in a pair in a function mandates a particular b, the reverse is not also true (in other words, while every x in a domain must have a unique associated y value, each y value in the range does not need a unique associated x value). In still other words, you can think of a function as a list of pairs saying, "For this a, this b.", but it doesn't necessarily say the same about "For this b..."
*The x coordinates that are part of the set we're checking on are the alleged function's domain, I'm pretty sure.
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... Damn that's a lot of text. Hopefully accurate?
EDIT: Also, hah, writing this up took exactly the amount of time I'd allotted myself before bed. Victory.
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Topic: [MILK] There were 12 eggs here what did you do with them? (Happy thread?!) (Read 16177998 times)